You are a secret agent and find that you have been pushed out of an airplane without a parachute. Fortunately, you are wearing a large overcoat (as secret agents often do). Thinking quickly, you are able to spread out and hold the overcoat so that you increase your overall area by a factor of two.

If your terminal velocity would be 45 m/s without the coat, what is your new terminal velocity with the coat?

The aerodynaminc drag force equals weight at the terminal velocity. Drag force is proportional to the square of the velocity, and the first power of the area. If area doubles, the velocity decreases by a factor of 1/sqrt2 = 0.707, to keep the drag force the same. The new terminal velocity is 45/sqrt2 = 31.8 m/s

To find your new terminal velocity with the coat, we need to consider the relationship between surface area and terminal velocity. Terminal velocity is the constant velocity that an object in freefall eventually reaches when the drag force on it equals its weight.

Given that you have spread out and doubled your area by using the overcoat, we can assume that the drag force acting on you will also double. According to the drag equation, the drag force is proportional to the square of the velocity and the surface area:

Drag force = 0.5 * air density * velocity^2 * surface area

Since the surface area has doubled and the drag force is directly proportional to the surface area, the drag force will also double.

However, the weight acting on you remains the same as before because you haven't changed your mass or the force of gravity. Therefore, when the drag force doubles, it will eventually equal your weight at a new terminal velocity.

Mathematically, we can express this as:

0.5 * air density * velocity_new^2 * 2 * area = weight

Simplifying the equation, we get:

air density * velocity_new^2 = weight / (2 * area)

Now we can rearrange the equation to solve for velocity_new:

velocity_new^2 = (weight / (2 * air density * area))

Finally, taking the square root of both sides, we find:

velocity_new = √(weight / (2 * air density * area))

Substituting the given values:

velocity_new = √(weight / (2 * air density * area)) = √(weight / (2 * air density * 2))

Using the given terminal velocity of 45 m/s without the coat, we can plug in the values and calculate the new terminal velocity:

velocity_new = √(45 / (2 * air density * 2))

Note: The calculation requires the density of the surrounding air, which is typically around 1.2 kg/m³ at sea level.

With this information, we can calculate the new terminal velocity.